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Haar wavelets, fluctuations and structure functions:

convenient choices for geophysics

S. Lovejoy,

Physics, McGill University, 3600 University st.,

Montreal, Que. H3A 2T8, Canada

lovejoy@physics.mcgill.ca

D. Schertzer,

LEESU, École des Ponts ParisTech, Université Paris-Est, 6-8 Av. B. Pascal,

77455 Marne-la-Vallée Cedex 2, France

Daniel.Schertzer@enpc.fr

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Abstract:

Geophysical processes are typically variable over huge ranges of space-time

scales. This has lead to the development of many techniques for decomposing series and

fields into fluctuations Δv at well defined scales. Classically, one defines fluctuations as

differences: (Δv)diff = v(x+Δx)-v(x) and this is adequate for many applications (Δx is the

“lag”). However, if over a range one has scalingΔv ∝ ΔxH , these difference fluctuations

are only adequate when 0<H<1, hence the need for other types of fluctuations. In

particular, atmospheric processes in the range ≈ 10 days to 10 years generally have -1 <

H < 0 so that a definition valid over the range -1< H <1 would be very useful for

atmospheric applications.

The general framework for defining fluctuations is wavelets. However the

generality of wavelets often leads to fairly arbitrary choices of “mother wavelet” and the

resulting wavelet coefficients may be difficult to interpret. In this paper we argue that a

good choice is provided by the (historically) first wavelet, the Haar wavelet (Haar,

1910) which is easy to interpret and - if needed - to generalize, yet has rarely been used

in geophysics. It is also easy to implement numerically: the Haar fluctuation (Δv)Haar at

lag Δx is simply proportional to the difference of the mean from x to x+Δx/2 and from

x+Δx/2 to x+Δx.

Using numerical multifractal simulations, we show that it is quite accurate, and

we compare and contrast it with another similar technique, Detrended Fluctuation

Analysis. We find that for estimating scaling exponents, the two methods are very

similar, yet Haar based methods have the advantage of being numerically faster,

theoretically simpler and physically easier to interpret.

1. Introduction:

In many branches of science - and especially in the geosciences - one commonly

decomposes space-time signals into components with well-defined space-time scales. A

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widely used method is Fourier analysis which - when combined with statistics - can be

used to quantify the contribution to the variance of structures of a given frequency and/or

wavenumber to the total variance of the process. Often however, real space analyses are

preferable, not only because they are simpler to perform, but more importantly - because

they are simpler to interpret. The latter are based on various definitions of fluctuations at

a given scale and location; the simplest being the differences:

Δv Δx( )( )diff ≡ δΔxv ; δΔxv = v x + Δx( )− v x( ) (1)

where the position is x, and the scale (“lag”) Δx and δ is the difference operator (the

parentheses with the “diff” subscript is only used when it is important to distinguish the

difference fluctuation from others). For simplicity, we consider a scalar function (v) of a

scalar position and/or of time. We consider processes whose statistics are translationally

invariant (statistically homogeneous in space, stationary in time). The resulting

fluctuations can then be statistically characterized, the standard method being via the

moments <Δvq>. When q = 2, these are the classical structure functions, when q ≠2, they

are the “generalized” counterparts.

The space-time variability of natural systems, can often be broken up into various

“scaling ranges” over which the fluctuations vary in a power law manner with respect to

scale. Over these ranges, the fluctuations follow:

Δv = ϕΔvΔxH (2)

where ϕΔx is a random field (or series, or transects thereof at resolution Δx). Below for

simplicity, we consider spatial transects with lag Δx, but the results are identical for

temporal lags Δt. In fluid systems such as the atmosphere, ϕ is a turbulent flux. For

example, the Kolmogorov law corresponds to taking v as a velocity component and

ϕ = ε1/3 with ε as the energy flux. The classical turbulence laws assumed fairly uniform

fluxes, typically that ϕ is a quasi Gaussian process.

Taking the qth power of eq. 2 and ensemble averaging, we see that the statistical

characterization of the fluctuations in terms of generalized structure functions is:

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Δv Δx( )q = ϕΔxq ΔxqH ≈ Δxξ q( ); ϕΔx

q = LΔx

⎛⎝⎜

⎞⎠⎟K q( ); ξ q( ) = qH − K q( )

(3) where ξ(q) is the structure function, K(q) is the moment scaling function that

characterizes the flux and L is that outer scale of the scaling regime (e.g. of a cascade

process). In the classical quasi-Gaussian case, K(q) = 0 so that ξ(q) is linear. More

generally, if the field is intermittent - for example if it is the result of a multifractal

process - then the exponent K(q) is generally nonzero and convex and characterizes the

intermittency. Since the ensemble mean of the flux spatially averaged at any scale is the

same (i.e. < φΔx > is constant, independent of Δx), we have K(1) = 0 and ξ(1) = H. The

physical significance of H is thus that it determines the rate at which mean fluctuations

grow (H>0) or decrease (H<0) with scale Δx.

There is a simple and useful expression for the exponent β of the power spectrum

E(k)≈ k-β. It is β =1+ξ(2) = 1+2H-K(2) which is a consequence of the fact that the

spectrum is a second order moment (the Fourier transform of the autocorrelation

function). K(2) is therefore often regarded as the spectral “intermittency correction”.

Note however that even if K(2) is not so large (it is typically in the range 0.1-0.2 for

turbulent fields), that the intermittency is nevertheless important: first because K(q)

typically grows rapidly with q so that the high order moments (characterizing the

extremes) can be very large, secondly because it is an exponent so that even when it is

small, whenever the range of scales over which it acts is large enough, its effect can be

large; in geo-systems the scale range can be 109 or larger.

Over the last thirty years, structure functions based on differences and with

concave rather than linear ξ(q) have been successfully applied to many geophysical

processes; this is especially true for atmospheric processes at time scales shorter than ≈

10 days which usually have 0 > H > 1. However, it turns out that the use of differences

to define the fluctuations is overly (and needlessly) restrictive. The problem becomes

evident when we consider that the mean difference cannot decrease with increasing Δx:

for processes with H<0, the differences simply converge to a spurious constant depending

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on the highest frequencies available in the sample. Similarly differences cannot grow

faster than Δx – they “saturate” at a large value - independent of Δx which depends on the

lowest frequencies present in the sample, hence they cannot be used when H >1. Overall,

whenever H is outside the range 0>H>1, the exponent ξ(q) is no longer correctly

estimated. The problem is that we need a definition of fluctuations such that Δv(Δx) is

dominated by wavenumbers in the neighbourhood of Δx-1. Appendix A gives more

details about the relation between real and Fourier space and the ranges of H relevant for

various types of fluctuation and fig. 1 shows the shape of typical sample series as H

varies.

The need to define fluctuations more flexibly motivated the development of

wavelets (e.g. (Holschneider, 1995), (Torrence and Compo, 1998)), the classical

difference fluctuation being only a special case, the “poor man’s wavelet”. To change the

range of H over which fluctuations are usefully defined one must change the shape of the

defining wavelet. In the usual wavelet framework, this is done by modifying the “mother

wavelet” directly e.g. by choosing various derivatives of the Gaussian (e.g. the second

derivative “Mexican hat”). Following this, the fluctuations are calculated as convolutions

with Fast Fourier (or equivalent) numerical techniques.

A problem with this usual wavelet implementation is that not only are the

convolutions numerically cumbersome, but the physical interpretation of the fluctuations

is largely lost. In contrast, when 0<H<1, the difference structure function gives direct

information on the typical difference (q = 1) and typical variations around this difference

(q = 2) and even typical skewness (q = 3) or typical Kurtosis (q = 4) or - if the probability

tail is algebraic – of the divergence of high order moments of differences. Similarly,

when -1 < H < 0 one can define the “tendency structure function” (below) which directly

quantifies the fluctuation’s deviation from zero and whose exponent characterizes the rate

at which the deviations decrease when we average to larger and larger scales. These poor

man’s and tendency fluctuations are also very easy to directly estimate from series with

uniformly spaced data and - with straightforward modifications - to irregularly spaced

data.

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In (Lovejoy and Schertzer, 2012a), (Lovejoy and Schertzer, 2012b) it is

argued that with only a few exceptions, for atmospheric processes in the “weather”

regime (Δt <≈10 days), that 0 < H <1 so that fluctuations increase with scale and the

classical use of differences is adequate. However, in the low frequency weather regime

(10 days ≈< Δt <≈ 10- 30 years), on the contrary, quite generally (in time, but not in

space!) we find -1< H < 0 so that fluctuations decrease with scale (although for larger Δt -

the “climate” regime – it is again in the range 0 > H > 1). In order to cover the whole

range (to even millions of years) one must thus use a definition of fluctuations which is

valid over the range 1> H > -1. This range is also appropriate for many solid earth

applications, including topography, geogravity and geomagnetism, for a review see

(Lovejoy and Schertzer, 2007). The purpose of this paper is to argue that the Haar

wavelet is particularly convenient, combining ease of calculation and of interpretation

and - if needed - it can easily be generalized to accommodate processes with H outside

this range. We demonstrate this using multifractal simulations with various H exponents

to systematically compare it to conventional spectral analysis and to another common –

and quite similar - analysis technique, the Detrended Fluctuation Analysis (DFA) method

(Peng et al., 1994), and its extension, the MultiFractal Detrended Fluctuation Analysis

(MFDFA) method (Kantelhardt et al., 2001), (Kantelhardt et al., 2002).

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Fig. 1: This shows samples of the simulations analysed in section 5. From bottom to top,

H = -7/10, -3/10, 3/10, 7/10. All have C1 = 0.1, α =1.8, the sections are each 213 points

long. One can clearly see the change in character of the series when H changes sign, at

the bottom, fluctuations tend to cancel out, fluctuations are “stable”, at the top, they tend

to reinforce each other, fluctuations are “unstable”.

2. Defining fluctuations using wavelets

Our goal is to find a convenient definition of fluctuations valid at least over the

range -1<H<1, but before doing so, it will be useful to first define “tendency fluctuations

Δv Δx( )( )tend . The first step is to remove the overall mean ( v x( ) ) of the series:

′v x( ) = v x( )− v x( ) and then take averages over lag Δx. For this purpose, we introduce

the operator T Δx defining the tendency fluctuation Δv Δx( )( )tend as:

Δv Δx( )( )tend =T Δxv =1Δx

′v ′x( )x≤ ′x ≤x+Δx∑ (4)

or, with the help of the summation operator S , equivalently by:

Δv Δx( )( )tend =

1Δx

δΔxS v ' ; S v ' = ′v ′x( )′x ≤x∑ (5)

we can also use the suggestive notation Δv Δx( )( )tend = Δv , see the schematic fig. 2.

When -1<H<0, Δv Δx( )( )tend has a straightforward interpretation in terms of the mean

tendency of the data to decrease with averaging. The tendency fluctuation is also easy to

implement: simply remove the overall mean and then take the mean over intervals Δx:

this is equivalent to taking the mean of the differences of the running sum.

It is now straightforward to define the Haar fluctuation by taking the second

differences of the mean:

Δv Δx( )( )Haar = H Δxv =2Δx

δΔx/22 S v = 2

Δxs x( ) + s x + Δx( )( )− 2s x + Δx / 2( )( )

= 2Δx

v ′x( )x+Δx/2< ′x <x+Δx

∑ − v ′x( )x< ′x <x+Δx/2∑⎡

⎣⎢⎤⎦⎥; s x( ) = S v

(6)

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Where H Δx is the Haar operator. Note the use of the shorthand notation s x( ) = S v . As

discussed in section 3, this is still a valid wavelet (see fig. 3), however it is almost trivial

to calculate and it can be used for -1<H<1 (appendix A; note there is an extra factor of

Δx-1 with respect to the Haar wavelet coefficient; see below). The numerical factor (2) is

used so that the Haar fluctuation is close to the usual differences when 0<H<1, (below).

While this may still look a little complicated it is actually quite simple: in words, the

Haar fluctuation at lag Δx is simply twice the difference of the mean from x to x+Δx/2

and from x+Δx/2 to x+Δx.

From the definitions, it is easy to obtain the relation:

H Δx = 2T Δx/2δΔx/2 = 2δΔx/2T Δx/2 (7) This is useful for interpreting the Haar fluctuations in terms of differences and tendencies

(section 5.2). Since the difference operator removes any constants, the tendency operator

can be replaced by an averaging operator and is thus insensitive to the addition of

constants; eq. 7 therefore means that the Haar fluctuation is simply the difference of the

series that has been degraded in resolution by averaging.

If needed, the Haar fluctuations can easily be generalized to higher order n by using

nth order differences on the sum s:

H Δx

n( )v =n +1( )Δx

δΔx / n+1( )n+1 s

(8) We note that the nth order Haar fluctuation is valid over the range -1<H<n; due to the

n+1th order difference, it is insensitive to polynomials of order n in the sum and of order

n-1 in the original series, although as we see below, for analyzing scaling series this

insensitivity has no particular advantage. Note further that H Δx1( ) = H Δx and

H Δx0( ) ′v =T Δx ′v hence when applied to a series with zero mean, H Δx

0( ) =T Δx .

Consider for a moment the case n = 2 which defines the “quadratic Haar

fluctuation”:

Δv Δx( )( )Haarquad =3Δx

s x + Δx( ) − 3s x + Δx / 3( ) + 3s x − Δx / 3( ) − s x − Δx( )( ) (9)

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this fluctuation is sensitive to structures of size Δx-1 – and hence useful - over the range

-1<H<2 and it “filters” out polynomials of order 1 (lines); this is thus essentially

equivalent to the quadratic MFDFA (Multi Fractal Detrended Fluctuation Analysis,

section 4) technique that we numerically investigate below.

Fig. 2: The top shows extracts of a multifractal temperature simulation with H = 0.5, at

one hour resolution (in the weather regime), the bottom is the same simulation but at 16

day resolution (in the low frequency weather regime, H = -0.4). One can clearly see

their differing characters corresponding to H>0, H<0 respectively (see also fig. 1). These

simulations are used to illustrate the two different types of structure function needed

when H>0 (the usual) and H<0, the “tendency” structure function. Whereas the usual

structure function yields typical differences in T over an interval Δt, so that the mean is of

no consequence the tendency structure function uses the average of the field with the

mean removed. In both cases the q =2 moment was chosen because it is directly related

to the spectrum. Taking the square root is useful since the result is then a direct measure

of the “typical” fluctuations in units of K.

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3. Fluctuations and Wavelets

Let’s now put these different fluctuations into the framework of wavelet analysis

where one defines fluctuations with the help of a basic “mother wavelet” Ψ(x) and

performs the convolution:

Δv x,Δx( ) = v ′x( )∫ Ψ ′x − xΔx

⎛⎝⎜

⎞⎠⎟d ′x

(10)

where we have kept the notation Δv to indicate “fluctuation”. Δv defined this way is

called a “wavelet coefficient”. The basic “admissibility” condition on Ψ(x) (so that it is a

valid wavelet) is that it has zero mean. If we take the fluctuation Δv as the symmetric

(centred) difference Δv = v x + Δx / 2( ) − v x − Δx / 2( ) this is equivalent to using:

Ψ x( ) = δ x −1 / 2( ) − δ x +1 / 2( ) (11)

where “δ” is the Dirac delta function (eq. 1, see fig. 3; due to the statistical translational

invariance, the coordinate shift by ½ is immaterial). As discussed, this “poor man’s”

wavelet is adequate for many purposes. As the generality of the definition (eq. 10)

suggests, all kinds of special wavelets can be introduced; for example orthogonal

wavelets can be used which are convenient if one wishes to reconstruct the original

function from the fluctuations, or to define power spectra locally in x rather than globally,

(averaged over all x).

The other simple to interpret fluctuation, the “tendency fluctuation” (eq. 4) can

also be expressed in terms of wavelets:

Ψ x( ) = I −1/2,1/2[ ] x( ) −I −L /2,L /2[ ] x( )

L; L >> 1

(12)

where I is the indicator function:

I a,b[ ] x( ) = 1 a ≤ x ≤ b0 otherwise (13)

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see fig. 3 for a schematic. The first term in eq. 12 represents the sum whereas the second

removes the mean where L>>1 is the overall length of the data set. The removal of the

mean in this way is necessary so that the wavelet satisfy the admissibility condition that

its mean is zero. The only essential difference between the wavelet defined by eq. 12 and

the “tendency fluctuation” defined in eq. 4 (where the mean is removed beforehand), is

the extra normalization by Δx in eq. 4 which changes the exponent by one.

Figure 3 also shows the wavelet corresponding to the Haar fluctuation:

Ψ x( ) =1; 0 ≤ x <1/ 2

−1; −1/ 2 ≤ x < 0

0; otherwise

(14)

To within the division by Δx, and a factor of 2, the result is the equivalent of using the

Haar fluctuations (eq. 6) as indicated in fig. 3. It can be checked (appendix A) that the

Fourier transform of this wavelet is ∝ k−1 sin2 k / 4( ) so that compared to the difference

and tendency flucutations whose mother wavelet has cutoffs only at high or low

wavenumbers, it is better localized in Fourier space with both low and high wavenumber

fall-offs (≈ k and ≈ k-1 respectively). The Haar fluctuation is a special case of the

Daubechies family of wavelets (see e.g. (Holschneider, 1995) and for some applications,

see (Koscielny-Bunde et al., 1998; Koscielny-Bunde et al., 2006), (Ashok et al.,

2010)).

If we are interested in fluctuations valid up until H = 2, we need merely take

derivatives. For example, the second finite difference wavelet is:

Ψ x( ) = 12

δ x +12

⎛⎝⎜

⎞⎠⎟+ δ x −

12

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟− δ x( )

(15)

(corresponding to Δv = (v(x+Δx)+v(x-Δx/2))/2 - v(x) using centred differences, fig. 4).

Alternatively, we can compare this with the quadratic Haar wavelet, eq. 9:

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−1/ 3; −1< x < −1/ 3ψ x( ) = 2 / 3; −1/ 3< x <1/ 3

1/ 3;0;

1 / 3< x <1otherwise

(16)

which is very close to the Mexican hat wavelet (see fig. 4) but which is numerically much

easier to implement. As with the Mexican hat, it is valid for -1<H<2. It turns out that for

analyzing multifractals, that even over their common range of validity (0<H<1), the Haar

structure function is somewhat better than the poor man’s wavelet based structure

function (i.e. the usual structure function). This is demonstrated on numerical examples

in section 5. Just as one can use higher and higher order finite differences to extend the

range of H values, so one can simply use higher and higher order derivatives of the

Mexican hat.

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Fig. 3: This shows the “poor man’s wavelet” by black bars representing the amplitudes

of Dirac δ functions, (the basis of the usual difference structure function, valid for 0<H<1,

this is only symbolic since the true δ function amplitude is of course infinite), the Haar

wavelet (the basis of the Haar structure function, uniform blue shading, the second

difference of the running sum, valid for -1<H<1), and the wavelet used for the “tendency”

structure function valid for -1<H<0), stippled shading.

Fig. 4: The popular Mexican hat wavelet (the second derivative of the Gaussian red,

valid for -1<H<2) compared with the (negative) second finite difference wavelet (black

bars representing the relative weights of δ functions, valid for 0<H<2)), and the second

order “quadratic” Haar wavelet (blue) eq. 9.

Wavelet mathematics are seductive - and there certainly exist areas such as speech

recognition - where both wavenumber and spatial localization of statistics are necessary.

However, the use of wavelets in geophysics is often justified by the existence of strong

localized structures that are cited as evidence that the underlying process is statistically

nonstationary/ statistically inhomogeneous (i.e. in time or in space respectively).

However, multifractal cascades produce exactly such structures - the “ singularities” - but

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their statistics are strictly translationally invariant. In this case the structures are simply

the result of the strong singularities but nevertheless, the local statistics are usually

uninteresting, and we usually average over them in order to improve statistical estimates.

Before continuing to discuss other related methods for defining fluctuations, we

should mention that there have been strong claims that wavelets are indispensable for

analyzing multifractals, e.g. (Arneodo et al., 1999). In as much as the classical

definition of fluctuations as first differences is already a wavelet, this is true (see however

the DFA discussion below). However, for most applications, one doesn’t need the full

arsenal of wavelet techniques. At a more fundamental level however, the claim is at least

debatable since mathematically, wavelet analysis is a species of functional analysis, i.e.

its objects are mathematical functions defined at mathematical points. On the contrary,

the generic multifractal process – cascades - are rather densities of singular measures so

that strictly speaking, they are outside the scope of wavelet analysis. Although in

practice the multifractals are always cut-off at an inner scale, when used as a theoretical,

mathematical tool for multifractal analysis, wavelets may therefore not always be

appropriate. A particular example where wavelets may be misleading is the popular

“modulus maximum” technique where one attempts to localize the singularities (Bacry et

al., 1989), (Mallat and Hwang, 1992). If the method is applied to a cascade process then

as one increases the resolution, the singularity localization never converges implying that

the significance of the method is not as obvious as is claimed.

4: Fluctuations as deviations from polynomial regressions:

the DFA, MFDFA techniques

We now discuss a variant method that defines the fluctuations in a different way,

but still quantifies the statistics in the manner of the generalized structure function by

assuming that the fluctuations are stationary; “MultiFractal Detrended Fluctuation

Analysis”. The MFDFA technique (Kantelhardt et al., 2001), (Kantelhardt et al., 2002)

is a straightforward generalization of the original “Detrended Fluctuation Analysis”

(DFA, (Peng et al., 1994)). The method works for 1-D sections, so consider the

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transect v(x) of series on a regular grid with resolution = 1 unit, N units long. The

MFDFA starts by replacing the original series by the running sum:

s x( ) = S v ' = ′v ′x( )

′x ≤x∑ (17)

(c.f. eq. 5). As discussed above, a sum is a finite difference of order -1 so that analysing s

rather than v allows the treatment of transects with H down to -1 (we will discuss the

upper bound shortly). We now divide the range into NΔx = int(N/Δx) disjoint intervals,

each indexed by i = 1, 2, …, NΔx (“int” means “integer part”). For each interval starting at

x = iΔx, one defines the “nth order” fluctuation by the standard deviation σs of the

difference of s with respect to a polynomial Fn(x = iΔx,Δx) = σs in the running sum of v

as follows:

Fn x = iΔx,Δx( ) = σ s =1Δx

s i −1( )Δx + j( )− pn,i j( )( )2j=1

Δx

∑⎡

⎣⎢

⎤

⎦⎥

1/2

; Δv Δx( )( )MFDFA =FnΔx

= σ s

Δx

(18)

where pn,ν(x) is the nth order polynomial regression of s(x) over the ith interval of length Δx.

It is the polynomial that “detrends” the running sum s; the fluctuation σs is the root mean

square deviation from the regression. F is the standard DFA notation for the “fluctuation

function”, as indicated, it is in fact a fluctuation in the integrated series quantified by the

standard deviation σs. Using F we have Fn= σs = (Δv)MFDFAΔx so in terms of the usually

cited DFA fluctuation exponent α (not the Levy α!) we have Fn ≈ Δxα and α =1+H so that

the (second order) spectral exponent β = 2α-1-K(2), so that the oft-cited relation β = 2α-1

ignores the intermittency correction K(2). The fluctuation (Δv) MFDFA is very close to the

nth order finite difference of v, it is also close to a wavelet based fluctuation where the

wavelet is of nth order, although this is not completely rigorous (see however

(Kantelhardt et al., 2001; Taqqu et al., 1995)). As a practical matter, in the above sum,

we start with intervals of length Δx = n+1 and normalize by the number of degrees of

freedom of the regression, i.e. by Δx-n not by Δx; formula 18 is the approximation for

large Δx.

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In the usual presentation of the MFDFA, one considers only a single realization of

the process and averages powers of the fluctuations over all the disjoint intervals i.

However, more generally, we can consider a statistical ensemble and average over all the

intervals and realizations. If in addition, we consider moments other than q = 2, then we

obtain the MultiFractal DFA (MFDFA) with the following statistics:

σ s Δx( ) q1/q

= Δx Δv Δx( )( )MFDFAq 1/q

= Δxh q( ); h(q) = 1+ ξ(q) / q (19)

where we have used the exponent h(q) as defined in (Davis et al., 1996) and

(Kantelhardt et al., 2002) and the relation Δv( )MFDFAq ≈ Δxξ q( ) . In what follows, we will

refer to the method as the MFDFA technique even though the only difference with

respect to the DFA is the consideration of the q ≠ 2 statistics.

From our analysis above, we see that the nth order MFDFA exponent h(q) is

related to the usual exponents by:

h q( ) = 1+ ξ q( ) / q = 1+ H( )− K q( ) / q; −1< H < n (20)

Note that in eq. 20, the 1 appears because of the initial integration of order 1 in the

MFDFA recipe, the method works up to H = n because the fluctuations are defined as the

residues with respect to nth order polynomials of the sum s corresponding to n+1th order

differences in s and hence n-1th order differences in v). The justification for defining the

MFDFA exponent h(q) in this way is that in the absence of intermittency (i.e. if K(q) =

constant = K(0) = 0), one obtains h(q) = H = constant. Unfortunately - contrary to what is

often claimed - the nonconstancy of h(q) - neither implies that that the process is

multifractal nor conversely does its constancy imply a monofractal process. To see this,

it suffices to consider the (possibly fractionally integrated, order H) monofractal β model

which has K(q) = C1(q-1) so that h(q) = 1+H- C1+C1/q which is not only non constant but

even diverges as q approaches zero (indeed the same q = 0 divergence occurs for any

universal multifractal with α<1, see eq. 21 below).

During the last ten years, the MFDFA technique has been applied frequently to

atmospheric data, of special note is the work on climate by A. Bunde and co-workers (e.g.

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(Koscielny-Bunde et al., 1998), (Kantelhardt et al., 2001), (Bunde et al., 2002), (Bunde

et al., 2005), (Lennartz and Bunde, 2009)). However, the exact status of the MFDFA

method is not clear since a completely rigorous mathematical analysis has not been done

(especially in the multifractal case), and the literature suffers from misleading claims to

the effect that the MFDFA is necessary because by its construction it removes

nonstationarities due to trends (linear, quadratic etc. up to polynomials order n-1) in the

data. While it is true that it does remove polynomial trends, these only account for fairly

trivial types of nonstationarity: beyond this, the method continues to make the standard

(and strong) stationarity assumptions about the statistics of the deviations which are left

over after performing this polynomial detrending. In particular it does nothing to remove

the most common genuine type of statistical nonlinearity in the geosciences: the diurnal

and annual cycles which still strongly break the scaling of the MFDFA statistics, and this

for any order n. Furthermore, the corresponding wavelet or finite difference definition of

fluctuations can equally well take into account these polynomial trends, and finally, the

method removes trends at all scales and locations so that this emphasis on detrending is

misleading. In other words, the MFDFA is a variant with respect to some of the wavelet

methods, in particular with respect to the Haar wavelet and its generalizations. While it

may indeed yield accurate exponent estimates it has the disadvantage that the

interpretation of the fluctuations is not straightforward. In comparison, the usual

difference and tendency fluctuations and structure functions have simple physical

interpretations in terms of magnitudes of changes (when 1>H>0) and magnitudes of

average tendencies (when -1<H<0).

5. A comparison of different fluctuations using multifractal

simulations

5.1 The simulations:

To have a clearer idea of the limitations of the various fluctuations for determining

the statistics of scaling functions, we shall numerically compare their performance and

their corresponding structure functions when applied to the characterization of

18

multifractals. In order to easily compare them with standard spectral analysis, we will

only consider the second order (q = 2) structure functions and will use simulations of

universal multifractals. Recall that universal multifractals are the result of stable,

attractive multiplicative cascade processes and have:

K q( ) = C1α −1

qα − q( ) (21)

where 0≤C1 is the codimension of the mean and 0≤α≤2 is the multifractality index (the

Lévy index of the generator). The limit α->1 gives K(q) = C1qlogq and may be obtained

from eq. 21 by using l’Hopital’s rule.

The following tests were all made on simulations with parameters α=1.8, C1 = 0.1

with H in the range -7/10<H<7/10 (typical parameters in meteorology and in solid earth

geophysics, see the reviews (Lovejoy and Schertzer, 2010b), (Lovejoy and Schertzer,

2007)). These parameters yield an intermittency correction K(2) = 0.18; 50 simulations

were averaged to estimate the ensemble statistics. The simulations were made using the

technique described in (Schertzer and Lovejoy, 1987) and refined in (Lovejoy and

Schertzer, 2010a) for the flux ϕ (i.e. H = 0). For the second fractional integration (to

obtain the field v from ϕ), we used a Fourier space power law filter k-H. For this

fractional integration, exponential cutoffs were used at high wavenbumbers to avoid

small scale numerical instabilities (this is standard for differentiation, i.e. when filtering

by k-H with H <0, it is equivalent to using the “Paul” wavelet; see (Torrence and Compo,

1998)). The raw series were 216 in length and were then degraded in resolution by

factors of 4 to 214 to avoid residual finite size effects in the simulation at the smallest

scales. In order to avoid spurious correlations introduced by the periodicity of the

simulations, the series were split in half so that the largest scale was 213.

In fig. 1 we already showed samples of the simulations visually showing the effect

of increasing H from -7/10 to 7/10. In fig. 5, we show the compensated spectra (obtained

using standard Hann windows to avoid spectral leakage). As expected, we see that they

are nearly flat, although for the lowest and highest factors of about 100.5 ≈3 there were

more significant deviations; the slopes of the central portions were thus determined by

regression and the corresponding exponents are shown in fig. 6. In fig. 5, the spectra

19

were averaged over logarithmically spaced bins, 10 per order of magnitude (with the

exception of the lowest factor of 10 where all the values were used). We also performed

the regressions using log-log fits using all the Fourier components (rather than just the

averaged values) in the same central range, these gave virtually identical exponent

estimates (the mean absolute differences in the estimated spectral exponent β were ≈

±0.007), and were thus not shown. Similarly, if instead of using log-log linear regression,

we use nonlinear regressions over the same range we obtain identical exponents to within

±0.009, and again these estimates were too similar to be worth showing. From fig. 6,

one can see that there seems to be a residual small bias of about -0.04 the origin of which

is not clear.

Fig. 5: The compensated spectra for an ensemble of 50 realizations, 214 each, α = 1.8, C1

= 0.1, intermittency correction = K(2) = 0.18, with H increasing from top to bottom from

-7/10 to 7/10. The dashed horizontal line is the theoretical behaviour indicated over the

range used to estimate the exponent (i.e. the highest and lowest factor of 100.5 in

wavenumber has been dropped). Each curve was offset in the vertical for clarity.

20

Fig. 6: This shows the regression estimates of the compensated exponents for the spectra

(red), Haar structure function (q =2, green), quadratic, q =2 MFDFA (dashed), The usual

difference (poor man’s) structure function (q =2, blue, for H>0), and the tendency

structure function (q =2, same line, blue for H<0).

21

Fig. 7: This shows the compensated Haar structure functions

SHaar Δx( ) = ΔvHaar Δx( )21/2

(solid lines) and the second order MFDFA technique

(dashed lines), again for H = -7/10 (top) to H = 7/10 (bottom), every 1/5. Each curve was

offset in the vertical for clarity by ¼.

22

Fig. 8: This compares the compensated Haar structure function (thick), the difference

structure function (thin, below the axis, for H =3/10 (bottom) and 1/10 orange, second

up), and the tendency structure function (third from bottom, thin red, H = -1/10), and top,

thin green, H = -3/10. It can be seen that the standard difference structure function has

poor scaling for nearly two orders of magnitude when H =1/10, and one order of

magnitude for H =3/10 (see fig. 7 for quantitative estimates).

We next consider the Haar structure function which we compare to the second

order, quadratic MFDFA structure function analogue (based on the MFDFA fluctuation

(Δv)MFDFA = F/Δx where F is the usual MFDFA scaling function, see eq. 18), the former is

valid over the range -1<H<1, the latter over the range -1<H<2. We can see (fig. 7) that

the Haar structure function does an excellent job with overall bias mostly around +0.01 to

0.02, (fig. 6), and that the MFDFA method is nearly as good (overall bias ≈ -0.02 to -

0.04). It is interesting to compare this in the same figure with the results of the tendency

structure function (H<0) and the usual difference (poor man’s) structure function (H>0),

fig. 8. The tendency structure function turns out to be quite accurate, except near the

limiting value H = 0 with the large scales showing the largest deviations. In comparison,

for the difference structure function, the scaling is poorest at the smaller scales requiring

a range factor of ≈ 100 convergence for H = 1/10 and a factor ≈ 10 for H = 3/10. The

23

overall regression estimates (fig. 6) show that the biases for both increase near their

limiting values H = 0 to accuracies ≈ ±0.1 (note that they remain quite accurate if one

only makes regressions around the scaling part). This can be understood since the

process can be thought of as a statistical superposition of singularities of various orders

and only those singularities with large enough orders (difference structure functions) or

small enough orders (tendency structure functions) will not be affected by the theoretical

limit H = 0. The main advantage of the Haar structure function with respect to the usual

(tendency or difference) structure functions is precisely that it is valid over the whole

range -1< H <1 so that it is not sensitive to the H =0 limit.

Before proceeding, we could make a general practical remark about these real space

statistics: most lags Δx do not divide the length of the series (L) exactly so that there is a

“remainder” part. This is not important for the small Δx<<L, for each realization there

are many disjoint intervals of length Δx however when L/3≈<Δx<L the statistics can be

sensitive to this since from each realization there will only one or two segments and

hence poor statistics. A simple expedient is to simply repeat the analysis on the reversed

series and average the two results. This can indeed improve the statistics at large Δx

when only one or a small number of realizations are available; it has been done in the

analyses below.

5.2 The theoretical relation between poor man’s, tendency and

Haar fluctuations, hybrid structure functions

We have seen that although the difference and tendency structure functions have

the advantage of having simple interpretations in terms of respectively the average

changes in the value of the process and its mean value over an interval, this simple

interpretation is only valid over a limited range of H values. In comparison, the Haar and

MFDFA fluctuations give structure functions with valid scaling exponents over wider

ranges of H. But what about their interpretations? We now briefly show how to relate

the Haar, tendency and poor man’s structure functions thus giving it a simple

interpretation as well.

24

In order to see the connection between the fluctuations, we use the “saturation”

relations:

δΔxv=dCtendv; H < 0

T Δxv=dCdiff v; H > 0

(22)

where Ctend, Cdiff are proportionality constants and =d

indicates equality in the random

variables in the sense of probability distributions ( a=db if Pr(a>ζ) = Pr(b> ζ) where “Pr”

means “probability” and ζ is an arbitrary threshold). These relations were easily verified

numerically and arise for the reasons stated above; they simply mean that for series with

H<0, the differences are typically of the same order as the function itself and that for H>0,

the tendencies are of the same order: the fluctuations “saturate”. By using eqs. 7 and eq.

22 we now obtain:

H Δxv = 2T Δx /2δΔx /2v=d2T Δx /2v=

d′CtendT Δxv; H < 0

H Δxv = 2δΔx /2T Δx /2v=d2δΔx /2v=

d′CdiffδΔxv; H > 0

(23) where C’tend, C’diff are “calibration” constants (only a little different from the unprimed

quantities – they take into account the factor of two and the change from Δx/2 to Δx).

Taking the qth moments of both sides of eq. 23, we obtain the results for the various qth

order structure functions:

Δv( )Haarq

Δv( )diffq = ′Cdiff

q ; H > 0

Δv( )Haarq

Δv( )tendq = ′Ctend

q ; H < 0

(24) This shows that at least for scaling processes that the Haar structure functions will be the

same as the difference (H>0) and tendency structure functions (H<0), as long as these are

“calibrated” by determining C’diff, and C’tend. In other words, by comparing the Haar

structure functions with the usual structure functions we can develop useful correction

factors which will enable us to deduce the usual fluctuations given the Haar fluctuations.

25

Although the MFDFA fluctuations are not wavelets, at least for scaling processes, the

same basic argument applies, since the MFDFA and usual fluctuations scale with the

same exponents, they can only differ in their prefactors, the calibration constants.

To see how this works on a scaling process, we confined ourselves to considering

the root mean square (RMS) S(Δt) = <Δv2>1/2; see fig. 9 which show the ratio of the usual

S(Δt) to those of the RMS Haar and MFDFA fluctuations. We see that at small Δx’s, the

ratio stabilizes after a range of about a factor 10 – 20 in scale and that it is quite constant

up to the extreme factor of 3 or so (depending a little on the H value). The deviations are

largely due to the slower convergence of the usual RMS fluctuations to their asymptotic

scaling form. From the figure, we can see that the ratios C’Haar = S/SHaar (fig. 9 upper left)

and corresponding C’MFDFA = S/SMFDFA (fig. 9, upper right) are well defined in the central

region, and are near unity, more precisely in the range ½< C’Haar <2 for the most

commonly encountered range of H: -4/10<H<4/10. In comparison (fig. 9, bottom) over

the same range of H, we have 0.23> C’MFDFA >0.025 so that the MFDFA fluctuation is

quite far from the usual ones and varies strongly with H. For applications, one may use

the semi-empirical formulae C’Haar ≈ 1.1 e-1.65H and C’MFDFA ≈ 0.075e-2.75H which are quite

accurate over the entire range -7/10<H<7/10. Although these factors in principle allow

one to deduce the usual RMS structure function statistics from the MFDFA and Haar

structure functions this is only true in the scaling regime. Since the corrections for Haar

fluctuations are close to unity for many purposes they can be used directly.

26

Fig. 9: These graphs compare the ratios of RMS structure functions ( S Δx( ) = Δv21/2

)

for the simulations discussed in the text.

Upper left: This shows the ratio of the RMS usual structure function (i.e. difference

when H>0, tendency when H<0) to the Haar structure function for H =1/10, 3/10,.. 9/10

(thick, top to bottom), and -9/10, -7/10, … -1/10, (thin, top to bottom). The Haar

structure function has been multiplied by 2ξ(2)/2 to account for the difference in effective

resolution. The central flat region is where the scaling is accurate indicating a constant

ratio C’Haar = S/SHaar which is typically less than a factor of two.

Upper right: The same but for the ratio of the usual to MFDFA RMS structure functions

after the MFDFA was normalized by a factor of 16 and the resolution correction 4ξ(2)/2 (its

smallest scale is 4 pixels).

Bottom left: This shows the ratio of the MFDFA structure function to the Haar structure

function both with the normalization and resolution corrections indicated above (note the

monotonic ordering with respect to H which increases from top to bottom).

27

5.3: Hybrid fluctuations and structure functions: an empirical

example with both H<0, and H>0 regimes

For pure scaling functions, the difference (1>H>0) or tendency (-1<H<0) structure

functions are adequate, the real advantage of the Haar structure function is apparent for

functions with two or more scaling regimes one with H>0, one with H<0. Can we still

“calibrate” the Haar structure function so that the amplitude of typical fluctuations can

still be easily interpreted? For these, consider eqs. 23 which motivate the definition:

H hybrid ,Δxv = max δΔxv,T Δxv( )

(25) of a “hybrid” fluctuation as the maximum of the difference and tendency fluctuations; the

“hybrid structure function” is thus the maximum of the corresponding difference and

tendency structure functions and therefore has a straightforward interpretation. The

hybrid fluctuation is useful if a unique calibration constant Chybrid can be found such that:

H hybrid ,Δxv≈dChybridH Δxv (26)

To get an idea of how the different methods can work on real data, we consider

the example of the global monthly averaged surface temperature of the earth. This is

obviously highly significant for the climate but the instrumental temperature estimates

are only known over a small number of years.

For this purpose, we chose three different series whose period of overlap was

1881-2008; 129 years. These were the NOAA NCDC (National Climatic Data Center) merged land air and sea surface temperature dataset (abbreviated NOAA NCDC below,

from 1880 on a 5°x5° grid, see (Smith et al., 2008) for details), the NASA GISS

(Goddard Institute for Space Studies) data set (from 1880 on a 2°x2° (Hansen et al.,

2010) and the HadCRUT3 data set (from 1850 to 2010 on a 5° x5° grid). HadCRUT3 is

a merged product created out of the HadSST2 (Rayner et al., 2006) Sea Surface

Temperature (SST) data set and its companion data set CRUTEM3 of atmospheric

temperatures over land, see (Brohan et al., 2006). Both the NOAA NCDC and the

NASA GISS data were taken from http://www.esrl.noaa.gov/psd/; the others from

http://www.cru.uea.ac.uk/cru/data/temperature/. The NOAA NCDC and NASA GISS are

both heavily based on the Global Historical Climatology Network (Peterson and Vose,

28

1997), and have many similarities including the use of sophisticated statistical methods

to smooth and reduce noise. In contrast, the HadCRUTM3 data is less processed with

corresponding advantages and disadvantages.

Fig. 10 shows the comparison of the difference, tendency, hybrid and Haar RMS

structure functions; the latter increased by a factor Chybrid = 100.35 ≈ 2.2. It can be seen that

the hybrid structure function does extremely well; the deviation of the calibrated Haar

structure function from the hybrid one is ±14% over the entire range of near a factor 103

in time scale. This shows that to a good approximation, the Haar structure function can

preserve the simple interpretation of the difference and tendency structure functions: in

regions where the logarithmic slope is between -1 and 0, it approximates the tendency

structure function whereas in regions where the logarithmic slope is between 0 and 1, the

calibrated Haar structure function approximates the difference structure function.

Before embracing the Haar structure function let us consider it’s behaviour in the

presence of nonscaling perturbations; it is common to consider the sensitivity of

statistical scaling analyses to the presence of nonscaling external trends superposed on

the data which therefore break the overall scaling. Even when there is no reason to

suspect such trends, the desire to filter them out is commonly invoked to justify the use of

quadratic MFDFA or high order wavelet techniques that eliminate linear or higher order

polynomial trends. However, for this purpose, these techniques are not obviously

appropriate since on the one hand they only filter out polynomial trends (and not for

example the more geophysically relevant periodic trends), while on the other hand, even

for this, they are “overkill” since the trends they filter are filtered at all – not just the

largest scales. The drawback is that with these higher order fluctuations, we loose the

simplicity of interpretation of the Haar fluctuation while obtaining few advantages. Fig.

11 shows the usual (linear) Haar RMS structure function compared to the quadratic Haar

and quadratic MFDFA structure functions. It can be seen that while the latter two are

close to each other (after applying different calibration constants, see the figure caption),

and that the low and high frequency exponents are roughly the same, the transition point

has shifted by a factor of about 3 so that overall they are quite different from the Haar

structure function, it is therefore not possible to simultaneously calibrate the high and low

frequencies.

29

Fig. 10: A comparison of the different structure function analyses (root mean

square, RMS) applied to the ensemble of three monthly surface series discussed in

appendix 10 C (NASA GISS, NOAA CDC, HADCRUT3), each globally and annually

averaged, from 1881-2008 (1548 points each). The usual (difference, poor man’s)

structure function is shown (orange, lower left), the tendency structure function (purple,

lower right), the maximum of the two (“Hybrid”, thick, red), and the the Haar in blue (as

indicated); it has been increased by a factor C =100.35 = 2.2 and the RMS deviation with

respect to the hybrid is ±14%. Reference slopes with exponents ξ(2)/2 ≈ 0.4, -0.1 are

also shown (black corresponding to spectral exponents β = 1+ξ(2) = 1.8, 0.8 respectively).

In terms of difference fluctuations, we can use the global root mean square ΔT Δt( )21/2

annual structure functions (fitted for 129 yrs > Δt >10 yrs), obtaining ΔT Δt( )21/2

≈

0.08Δt 0.33 for the ensemble, in comparison, (Lovejoy and Schertzer, 1986) found the very

similar ΔT Δt( )21/2

≈ 0.077Δt0.4 using northern hemisphere data (these correspond to βc =

1.66, 1.8 respectively). Reproduced from (Lovejoy and Schertzer, 2012a).

30

Fig. 11: The same temperature data as fig. 10; a comparison of the RMS Haar structure

function (multiplied by 100.35 = 2.2), the RMS quadratic Haar (multiplied by 100.15 = 1.4)

and the RMS quadratic MFDFA (multiplied by 101.5 = 31.6). Reproduced from (Lovejoy

and Schertzer, 2012a).

If there are indeed external trends that perturb the scaling, these will only exist at

the largest scales; it is sufficient to remove a straight line (or if needed second, third or

higher order polynomials) from the entire data set i.e. at the largest scales only. Fig. 12

shows the result when the Haar structure function is applied to the data that has been

detrended in two slightly different ways: by removing a straight line through the first and

last points of the series and by removing a regression line. Since these changes

essentially effect the low frequencies only, they mostly affect the extreme factor of two in

scale. If we discount this extreme factor 2, then in the figure we see that there are again

two scaling regimes, the low frequency one is slightly displaced with respect to the Haar

structure function, but not nearly as much as for the quadratic Haar. In other words,

removing the linear trends at all scales as in the quadratic Haar or the quadratic MFDFA

is too strong to allow a simple interpretation of the result and it is unnecessary if one

wishes to eliminate external trends.

31

Fig. 12: The same temperature data as fig. 10; a comparison of the RMS Haar structure

function applied to the raw data, after removing linear trends in two ways. The first

defines the linear trend by the first and last points while the second uses a linear

regression; all were multiplied by Chybrid = 100.35 = 2.2.

In summary, if all that is required are the scaling exponents, the Haar structure

function and quadratic MFDFA seem to be a excellent techniques. However, the Haar

structure function has the advantage of numerical efficiency, simplicity of

implementation, simplicity of interpretation.

6. Conclusions:

Geosystems are commonly scaling over large ranges of scale in both space and time.

In order to characterize their properties, one decomposes them into elementary

components with well defined scales and characterizes the variability with the help of

various statistics. Their scaling behaviours can then be characterized by exponents

obtained by systematically changing scale. The classical analysis techniques are spectral

analysis (Fourier decomposition) and structure functions based fluctuations defined as

differences. However, over the last twenty years, the development of wavelets has

provided a systematic mathematical and fairly general basis for defining fluctuations. As

32

far as geophysical applications are concerned, this generality has not proved to be so

helpful. This is because different workers use different wavelets and these are often

chosen for various mathematical properties which are not necessarily important for

applications. In addition, the numerical determination of the wavelet coefficients is often

cumbersome. Finally, the interpretation of the corresponding wavelet coefficients is

often obscure and attention has been largely focused on the exponents.

These features of wavelet analysis have lead to the continued popularity of a

slightly different wavelet- like technique - the Detrended Fluctuation Analysis (DFA)

method and its extension, the MultiFractal DFA (MFDFA). An unfortunate consequence

is that the criterion for the applicability of the different techniques has largely been

ignored (the range of H exponents), and the advantages and disadvantages of different

approaches has not been critically (and quantitatively) evaluated. In this paper, we argue

that the historically first wavelet - the Haar wavelet - has several advantages. These are

not so much at the level of the accuracy of the exponents – which turn out to be about the

same as for spectral analysis and the DFA technique – but rather in the ease of

implementation and interpretation of the results. This is demonstrated both with the help

of multifractal simulations (exponents) but also the ease of interpretation with an

atmospheric example where we show that with “calibration” the Haar structure functions

can be made very close to both the usual difference structure functions (in regions where

1>H>0) and to simple “tendency” structure functions (in regions where -1<H<0). This

simplicity is already lost when using either the DFA or even higher order wavelet based

generalizations of Haar fluctuations. The Haar structure functions have recently been

applied systematically to atmospheric data where they have clearly shown that there are

not two regimes (weather and climate) but rather three (weather <≈10 days, low

frequency weather (between ≈10 days and ≈ 10-30 years and the climate ≈>10 - 30 years),

(Lovejoy and Schertzer, 2012a).

33

Appendix A: Relation between real and Fourier space

analyses and the limiting H exponents

We repeatedly underlined the limiting H values associated with specific type of

fluctuations. In this appendix we clarify the origin of these limits by considering the

relation between the fluctuations and the spectra.

Consider the statistically translationally invariant process v(x) in 1-D: the statistics

are thus independent of x and this implies that the Fourier components are “δ correlated”:

v k( )v ′k( ) = δk+ ′k v k( ) 2 ; v k( ) = e− ikxv x( )dx∫

(27)

If it is also scaling then the spectrum E(k) is a power law: E k( ) ≈ v k( ) 2 ≈ k−β

(where

here and below, we ignore constant terms such as factors of 2π etc.). Let us first consider

the difference fluctuation. In terms of its Fourier components, this fluctuation is thus:

Δv x,Δx( )( )diff = v x + Δx( )− v x( ) = eikx v k( ) eikΔx −1( )dk∫ (28)

so that the F.T. of (Δv(x,Δx))diff is v k( ) eikΔx −1( ) . We first consider the statistics of quasi-

Gaussian processes for which C1 = 0, ξ(q) = Hq. Assuming statistical translational

invariance, we drop the x dependence and relate the second order structure function to the

spectrum:

Δv Δx( ) 2 = 4 eikx v k( )2 sin2 kΔx

2⎛⎝⎜

⎞⎠⎟dk∫ ≈ eikxk−β sin2 kΔx

2⎛⎝⎜

⎞⎠⎟dk∫

(29)

As long as the integral on the right converges, then using the transformation of variables,

x→λx , k→λ−1k , we find:

Δv Δx( ) 2 ∝ Δxξ 2( ) ≈ eikxk−β sin2 kΔx2

⎛⎝⎜

⎞⎠⎟dk∫ ∝ Δxβ−1

(30)

34

so that β = ξ(2)+1 = 2H +1 (C1 = 0 here). However, for large k, the integrand ≈ k-β which

has a large wavenumber divergence whenever β<1. However, since for small k,

sin2 kΔx / 2( )∝ k2 , there will be a low wavenumber divergence only when β>3. In the

divergent cases, although real world (finite) data will not diverge, the structure functions

will no longer characterize the fluctuations, but will depend spuriously on the either the

highest or lowest wavenumbers present in the data. In the quasi Gaussian case, or when

C1 is small, we have ξ(2) ≈ 2H and we conclude that the using first order differences to

define the fluctuations leads to second order structure functions being meaningful in the

sense that they adequately characterize the fluctuations whenever 1<β<3, i.e. 0<H<1.

Since 0<H<1 is the usual range of geophysical H values, and the difference

fluctuations are very simple, they are commonly used. However, we can see that there

are limitations; in order to extend the range of H values, it suffices to define fluctuations

using finite differences of different orders. To see how this works, consider using second

(centred) differences:

Δv x,Δx( )( )second = v x( )− 12v x + Δx( )− v x − Δx( )( ) = eikx v k( ) 1− 1

2eikΔx/2 + e−ikΔx/2( )⎡

⎣⎢⎤⎦⎥dk∫

= 2 eikx v k( )sin kΔx4

⎛⎝⎜

⎞⎠⎟2

dk∫ (31)

repeating the above arguments we can see that the relation β = ξ(2)+1 holds now for

1<β<5, or (with the same approximation) 0<H<2. Similarly, by replacing the original

series by its running sum (a finite difference of order -1) – as done in the MFDFA

technique – we can extend the range of H values down to -1.

More generally, going beyond Gaussian processes we can consider intermittent

Fractionally Integrated Flux processes (section 5.1), which have v k( ) ≈ ε k( ) k −H

, we see

that the F.T. of (Δv(x,Δx))diff is eikΔx −1( ) k −H ε k( ) . This implies that for low

wavenumbers, v k( ) ≈ k 1−H ε k( ) (k<<1/Δx) whereas at high wavenumbers,

v k( ) ≈ k −H ε k( )

(k>>1/Δx) hence since the second characteristic function of ε(x) has

35

logarithmic divergences with scale for both large and small scales ( log ελq = K q( ) logλ ),

we see that for 0<H<1, that the fluctuations Δv(Δx) are dominated by wavenumbers k ≈

1/Δx, so that for this range of H, fluctuations defined as differences capture the variability

of Δx sized structures, not structures either much smaller or much larger than Δx. More

generally, since the F.T. of the nth derivative dnv/dxn is ik( )n v k( ) and the finite derivative

is the same for small k but “cut-off” at large k, we find that nth order fluctuations are

dominated by structures with k ≈1/Δx as long as 0 < H <n. This means that Δv(Δx) does

indeed reflect the Δx scale fluctuations.

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